## Abstract

We study theoretically the focusing effects when a Gaussian beam passing through a graded photonic crystal consisting of air holes with gradually varying radii. The device is used to couple the light between the conventional and photonic crystal waveguides. The coupling efficient is about 4 times higher than that by the linear taper structure.

©2006 Optical Society of America

## 1. Introduction

In recent years, photonic crystals (PCs) are attractive optical structures for controlling and manipulating the flow of light. Various devices such as beam splitters [1, 2], filters [3] and laser [4] have been proposed. Moreover, negative refraction [5–8] has been investigated in lens applications. Subwavelength focused spot size can be achieved. In Ref. 9, the PC lens formed by the periodically arranged rods to focus light is studied. The light with different polarizations can be focused at different positions to yield a polarization splitter. [9] PC lens formed by dielectric rods is also proposed by L. Sanchis *et al*. [10] By using the genetic algorithm to remove the arbitrary scatterers (dielectric rods), the optimized structure to focus light can be obtained.

In fact, PC structures formed by dielectric rods are difficult to use due to the lack of the vertical confinement of the light. The graded PC proposed by Emmanuel Centeno and David Cassagne [11] are formed by air holes. Gradual modifications of the lattice periodicity make it possible to bend the light at micrometer scale. The structure can provide a frequency-selective tunable bending for the application of wavelength division multiplexing. Recently, we have presented the preliminary results to understand the optical behavior in a graded PC. [12]. In this study, the graded PC are formed by air holes with gradually varying radii. The device can be used to focus a collimated light. Since the focused spot size is close to the lattice constant of the PC structure, the devices are useful to couple the light between the conventional and the PC waveguides.

## 2. Simulation and results

A two-dimensional (2-D) graded PC consisting of m × n air holes arranged in a squared lattice is illustrated in Fig. 1. The refractive index of the surrounding dielectric material is 3.46 (silicon). The center-to-center distance between the air holes is the lattice constant, *a*. For the optical devices operated at the wavelength of 1550nm, since the width of a conventional single-mode waveguide and optical fiber (around several micrometers) is around 25 times larger than that of the photonic crystal single-mode waveguides (several tenths of a micrometer), the value of n (number of columns of air holes) which defines the width of the graded PC is chosen to be 25 in this study. The value of m is varied from 1 to 14 layers.

During the fabrication of the PC structures by e-beam lithography, the radius of air holes is a critical parameter. The air holes with small radius may cause incomplete development of the photoresist and low aspect ratio during dry etching. For air hole with large radius, the horizontal undercut may cause the failure in dry etching process. In this study, the air-hole radii of the central column and the edge column of the graded PC are chosen to be r_{1}=0.27*a* and r_{13}=0.39*a*, respectively. The radii are linearly increased from the central to the edge column. A pulsed Gaussian beam with its width of 25*a* is launched from the bottom of the structure along the Z-axis. The normalized central frequency of the pulse is *f*=0.4 (*a*/λ) where λ is the wavelength. 2-D finite-difference time-domain (FDTD) method is utilized to study the propagation behaviors of the electromagnetic (EM) waves. The light source is H-polarized (so the electric field lies on the propagation plane).

Frequency analysis is utilized to study the focusing effect in the domain of the normalized frequency *f*=0.1~0.5. The corresponding wavelength is 2*a*~10*a*. The field intensities at the positions along Z-axis from (0, 6a) to (0, 130a) are analyzed. By discrete Fourier transform (DFT), the intensity spectra along Z-axis are obtained. Figure 2 shows the intensity spectra along Z-axis for different m values. These spectra have been normalized to that of the input light source. From these spectra, we can observe that the normalized frequency of the light passing through the graded PC with higher transmission is between 0.22 and 0.3, between 0.33 and 0.41 and between 0.45 and 0.50. Using planewave expansion method (PWE), we calculate the bandgap along Γ-X direction of the uniform (non-graded) PC structures formed by air holes in square lattice. The calculation is performed for the structures with the radius of the air holes from 0.27*a* to 0.39*a*. The result is shown in Fig. 3. In FDTD simulation, the center of the Gaussian beam where the intensity of the beam is maximum is aligned to the center of the graded PC where the radius is 0.27*a*. Since the maximum intensity of a Gaussian beam is in the center of the beam, the transmission spectra at the center of the graded PC where the radius is 0.27*a* may dominate the transmission rate of the graded PC. In Fig. 3, we observe that the allowed bands for the radius of 0.27*a* exist for the normalized frequency between 0.2 and 0.31, between 0.34 and 0.4, and above 0.44. These frequencies correspond to the frequencies of the higher transmission rate analyzed from Fig. 2. We can also observe that in Fig. 3, between the normalized frequency of 0.28 and 0.31, a complete allowed band exists for the radii from 0.27*a* to 0.39*a*. Consequently, we can observe the fact that for all cases in Fig. 2, the transmission is higher for the normalized frequency between 0.27 and 0.3. The maximum intensity is found around the normalized frequency of 0.275 for the case of m=10. Therefore, we choose the normalized frequency of 0.275 to study the focusing effect in different cases.

A continuous wave at the frequency *f*=0.275 is launched into the structure with different numbers of layers m to study the spot size, focus intensity, focal length (*F*), and numerical aperture (NA). The light source is a Gaussian beam with its width of 25*a*. The focal length is the distance between the center of the graded PC (Z=0) to the focus. The NA is defined as

where *R* and *n _{Si}* are the half width of the PC structure which is (m/2)×

*a*and the refractive index of silicon, respectively. The intensity at the focus and the spot size are shown in Fig. 4(a). The NA and the focal length are shown in Fig. 4(b).

We can observe that the spot size and the focal length decrease as the number of the layers (m) increases. The NA increases as m increases. These behaviors of increasing m are similar to increasing the central thickness or decreasing the curvature of the conventional lens in classical optics. From these viewpoints, the graded PC can be regarded as a lens. As m increases from 1 to 10, the intensity at focus increases. However, as m increases from 10 to 14, the intensity at focus decreases. Figure 5 shows the intensity distributions of magnetic field in steady state for the structure of m=10 and 14. We can observe that the diffracted light from the graded PC is denser for m=14 than m=10. The higher diffracted light leads to diminish the intensity at the focus. Therefore, the intensity at focus does not always increases as m increases from 1 to 14. The maximum intensity is found as m is 10.

In Fig. 4(a), we can observe that the minimum spot size is around 2*a*. This value is close to the mode width in PC waveguides formed by line defects. We place a PC waveguide with single line defect at the focus of the graded PC. We launch a Gaussian beam as described above to couple the light into the PC waveguide by using the graded PC with m=10. The propagation length from the bottom edge of the graded PC to the entrance of the PC waveguide is 32.6a. (The distance from the bottom edge of the graded PC to the focus.) A linear taper structure [13] with the identical propagation length is simulated as a reference. Figures 6(a) and 6(b) show the intensity distributions for the linear taper and the graded PC, respectively. The coupling efficiencies (transmitted power/ light source power) are around 15% and 60% in the linear taper case and the graded PC case, respectively. This result shows the advantage to use the graded PC for light coupling between the conventional and PC waveguides.

Furthermore, we also studied the effect of reducing the number of columns. Figure 6(c) shows the result of the graded PC in which the number of columns is reduced to be 9 and the number of layers is 10. We can observe that the depth of focus increases when the number of columns reduces. This behavior is similar to the phenomena in classical optics. The oblique incidence of the light source is also studied. The results of incident angle at 5, 10, and 15 degrees are shown in Figs. 7(a), 7(b), and 7(c), respectively. The intensity of the focus becomes gradually weaker as the incident angle increases. This is due to the stronger reflection from the PC structure at larger incident angle.

## 3. Discussion and conclusion

In this work, the graded PC is studied for focusing a collimated Gaussian beam. The design of the graded PC is proposed by using the frequency analysis. The high transmission region in the transmission spectra correspond to the allowed band of the uniform (non-graded) PC along the Γ-X direction. Since the time consumption of the calculation for PWE method is much lower than that of FDTD method. This result implies that PWE may be the suitable method to qualitatively predict the high transmission frequency region of the graded PC.

Taking fabrication limitation into account, the feature size of the air holes is restricted. To operate the graded PC at wavelength of 1550nm, the corresponding lattice constant *a* is 426nm. The diameter of the central air holes is 230nm. This feature size is approaching the minimum for e-beam lithography and dry etching technique to obtain a perfect round form. The diameter of the edge air holes is 332nm. The minimum width of the dielectric wall that separates two adjacent air holes is 94nm. This feature size is the minimum for the fabrication process to obtain the result similar to the design. Therefore, the minimum and the maximum diameters of the air holes in this study are fixed. The corresponding difference of hole diameters between two adjacent columns is only 8.6nm which approaches the resolution of the e-beam lithography. Therefore, the radius is changed linearly instead of non-linear variation. Additionally, the corresponding width of the graded PC is around 10μm. This value corresponds to the modal diameter of a single-mode optical fiber. This design will help the light coupling between the input fiber, the input silicon waveguide and the PC waveguide.

The light coupling effect using our structure (60%) is better than that using a linear taper (15%). This originates from the fact that the structures studied in Ref. 13 are formed by dielectric rods in air leading to excite much less modes in the air taper than in the dielectric taper. Since the high-order modes are excited in the dielectric linear taper [Fig. 6(a)], the light coupling is deteriorated. Additionally, the NA in Figs. 6(a) and 6(b) around 1.3 is much higher than the cases studied in Ref. 13 around 0.2. Therefore, our structure may be more suitable to be used in the high NA optical systems to serve as a solid immersion lens [14].

Equifrequency contours (EFC) can be used to predict the direction of a beam refracted from homogeneous medium into PCs formed by the periodically arranged uniform scatterers. [5, 6] In Ref. 11, the deflection of the light in the graded PC is also analyzed by EFC. The lattice variation of the graded PC is less than 1% of the lattice constant. The result shows that this approximation may be feasible. [11] We have also adopted EFC to analyze the deflection direction of the beam in the graded PC. However, the result does not show a good consistence with the simulation results obtained by the FDTD. The variation of the air hole size is around 7% of the lattice constant in our case. Therefore, this result implies that the EFC cannot be applied for the graded PC with the size variation of the scatterers larger than 7%.

It is well known that a PC can be treated as a homogeneous medium when the wavelength is very large compared to the lattice constant. [7, 9] However, in this study, since the wavelength of EM waves is only around 3.6*a*, the effective index method cannot be used to predict quantitatively the focusing properties (focal length, spot size, NA, etc.).

The result of the graded PC with reduced width shows that the depth of focus increase. Additionally, the spot size and the focal length of the graded PC decrease as the number of the layers (m) increases. These phenomena show that the graded PC can be regarded as a typical lens in which the central optical path is larger than that at the edge. Since the diameter of the air holes at the edge of the structure is larger than that at the center, we can qualitatively deduce the fact that the central optical path is longer than the edge optical path. Therefore, the graded PC can provide the focusing phenomena.

The optimized graded PC is m=10 at frequency *f*=0.275. The focal length, spot size, and NA are 28.6*a*, 1.94*a*, and 1.4, respectively. The coupling efficiency for the graded PC is around 4 times higher than the linear taper structure.

## References and links

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